Weak pseudocompactness on spaces of continuous functions

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• 作者：A. Dorantes-Aldama^a ; ^{alejandro_dorantes@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; R. Rojas-Herná ; ndez^b ; ^{satzchen@yahoo.com.mx" class="auth_mail" title="E-mail the corresponding author} ; Á ; . Tamariz-Mascarú ; a^c ; ^{atamariz@unam.mx" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 54C35 ; 54D35 ; 54C45 ; 91A44 ; secondary ; 54B10 ; 54D45
• 刊名：Topology and its Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：196
• 期：part_PA
• 页码：72-91
• 全文大小：515 K

文摘

A space X is weakly pseudocompact   if it is 243c212a04" title="Click to view the MathML source">G_δ$G_{\delta }$-dense in at least one of its compactifications. X has property  D_Y$D_Y$ if for every countable discrete and closed subset N of X  , every function f:N→Y$f:N\to Y$ can be continuously extended to a function over all of X. O-pseudocompleteness is the pseudocompleteness property defined by J.C. Oxtoby [17], and T-pseudocompleteness is the pseudocompleteness property defined by A.R. Todd [24].

In this paper we analyze when a space of continuous functions C_p(X,Y)$C_p(X,Y)$ is weakly pseudocompact where X and Y   are such that C_p(X,Y)$C_p(X,Y)$ is dense in Y^X$Y^X$. We prove: (1) For spaces X and Y such that X   has property D_Y$D_Y$ and Y   is first countable weakly pseudocompact and not countably compact, the following conditions are equivalent: (i) C_p(X,Y)$C_p(X,Y)$ is weakly pseudocompact, (ii) C_p(X,Y)$C_p(X,Y)$ is O-pseudocomplete, and (iii) C_p(X,Y)$C_p(X,Y)$ is T-pseudocomplete. (2) For a space X and a compact metrizable topological group G such that X   has property D_G$D_G$, the following statements are equivalent: (i) C_p(X,G)$C_p(X,G)$ is pseudocompact, (ii) C_p(X,G)$C_p(X,G)$ is weakly pseudocompact, (iii) C_p(X,G)$C_p(X,G)$ is T  -pseudocomplete, and (iv) C_p(X,G)$C_p(X,G)$ is O-pseudocomplete. (3) For every space X  , $C_p^{⁎}(X)$ and $C_p^{⁎}(X,\mathbb{Z})$ are not weakly pseudocompact. Throughout this study we also consider several completeness properties defined by topological games such as the Banach–Mazur game and the Choquet game.